In nuclear medicine, a radiopharmaceutical is injected into a patient and accumulates in tissues of physiological interest (e.g., cancer tumors or heart muscle). The gamma-ray emissions from this radiopharmaceutical are then observed and the local density of radiopharmaceutical within the patient reconstructed. Two clinical imaging techniques, single photon emission computed tomography (SPECT) and positron emission tomography (PET), are based on this simple idea and are currently available commercially. Both these imaging techniques use filtered backprojection (FBP) algorithms for the reconstruction of the three-dimensional (3D) distribution of radiopharmaceuticals within the patient. The advantages and limitations of SPECT and PET have been reviewed extensively in the literature and will not be further discussed here. It is generally acknowledged, however, that one of the major limitations of SPECT is the loss of counts due to the collimator in the gamma-ray camera. Typically only one gamma ray is observed for every ten thousand emissions, the vast majority of these gamma rays are absorbed in the collimator. In 1974 Todd, Nightingale, and Everett proposed a new type of gamma camera that would replace the collimator with a method of “electronic collimation” based on Compton scattering. Compton kinematics implies a relationship between the angle of scatter and the energy deposited in a detector. Knowledge of this energy and, therefore, the angle of scatter can be used to estimate the direction of an incident gamma-ray. The success of this strategy is notably demonstrated by the gamma-ray telescopes currently surveying the sky for sources of gamma emissions. In medical applications, however, the patient is located at a finite distance, so that the interpretation of the angular information provided by Compton events is complicated by huge parallax effects. In landmark research during the 1980s and 1990s, Manbir Singh and his group constructed and tested a prototype Compton camera. That research revealed both the promise and difficulties associated with Compton cameras. Many of the technical difficulties associated with Compton cameras (e.g., limited energy resolution, restrictive detector geometries, and events involving multiple scatters) are currently being remedied by advances in detector technology. Among the most significant of these problems is the 3D reconstruction of the radiopharmaceutical distribution from the data provided by a Compton camera.
The reconstruction of 3D images from Compton camera data poses a formidable mathematical problem. Unlike collimated cameras that reveal the direction of each gamma-ray, the incident direction of each event in a Compton camera is restricted to a cone. This geometric difference completely changes the nature of the reconstruction problem. In his original studies, Singh in collaboration with Hebert and Leahy implemented a maximum likelihood (ML) reconstruction (and tested other iterative methods). In recent years numerous alternative algorithms have been proposed and studied. Roughly speaking, these algorithms can be divided into iterative and analytic inversion methods. Both types of algorithm require the construction of a mathematical model that describes the Compton camera data in terms of the source distribution of radiopharmaceutical substance. Generally, the iterative methods can accommodate more intricate and accurate models, whereas, the analytic inversions require simpler models that are amenable to direct mathematical analysis. Most experimental groups working on Compton cameras have used iterative algorithms similar to the ML techniques of Singh et al. However, the reason for preferring iterative algorithms was not the accuracy of the mathematical model. After all, FBP algorithms (that are prototypes of analytic inversion) are generally used in clinical SPECT and PET despite the existence of ML algorithms for more accurate models. Basically, the iterative algorithms dominate Compton reconstruction because, until the late 1990s, no analytic algorithm existed for Compton cameras.
An algorithm for Compton cameras developed by Cree and Bones restricts the acceptable data as much as would a standard collimator—and is, therefore, unsuitable for practical applications. Shortly thereafter, Basko, Gullberg and Zeng produced and patented an algorithm (for the same mathematical model) that removed all restrictions on the Compton data. This algorithm explicitly rebins the Compton data into a form that leads to a 2D Radon transform (and, therefore, FBP). This rebinning procedure, which is the major drawback of the technique, overcomes the basic problem of Compton camera reconstruction; namely, the cone of incident gamma-ray directions. The rebinning implements a complex summation over spherical harmonics that arises from the cone of acceptance. Other researchers developed analytic-inversion algorithms that explicitly replace integrals over the acceptance cone with a summation over spherical harmonics. For example, Parra and Tomitani and Hirasawa retain the spherical harmonic expansions—truncating the series to a finite number of terms for numerical implementation. Although Parra and Tomitani and Hirasawa use a slightly different mathematical model than Basko et al, they face the same problem; namely, the additional degree of freedom introduced by the acceptance cone of the Compton camera. In reading the papers of Parra and Tomitani and Hirasawa, one is reminded of the initial papers of Cormack in which the Radon transform was rediscovered in terms of circular harmonics. One recalls that, when the problem was properly reformulated, the circular harmonics disappeared and the corresponding FBP algorithms emerged.
While these analytic algorithms were being developed, other groups examined alternative strategies. In the process of designing and building the C-SPRINT system, the group at the University of Michigan engaged in a broad-based campaign to establish Compton cameras as a viable imaging technology. Not only were simulations and sophisticated statistical analysis used for camera design, but the reconstruction problem was attacked using a variety of techniques. As a leading research institution in ML and statistical reconstruction, the University of Michigan implemented the most sophisticated iterative algorithms currently available. Furthermore, non-iterative reconstruction algorithms were also examined. Sauve et al studied matrix inversion techniques for a model of their system. The geometrical symmetries of the system were exploited in an attempt to handle the otherwise intractably large set of equations. In another approach, that is relevant to the research reported in this paper, Wilderman et al developed the software for list-mode back-projection of Compton scatter events into the appropriate cone geometries. In this research, they were joined by Rohe et al who also developed backprojection algorithms for the cone geometry of Compton cameras. Unfortunately, simple backprojection onto the appropriate cones (without filtering) does not produce accurate reconstruction of the source distribution.
With the current art, reconstruction of a radiopharmaceutical source distribution typically utilizes iterative algorithms. The current art also provides analytic-inversion techniques that may be inherently faster to execute but are typically deficient with respect to relative immunity to statistical noise and the desired accuracy of reconstructing the source distribution. Hence, there is a real need to develop methods and apparatus that utilize analytic-inversion techniques while providing accuracy in reconstructing a radiopharmaceutical source distribution within a patient in order to properly diagnose the patient.